## Abstract

We study global regularity properties of Sobolev homeomorphisms on n-dimensional Riemannian manifolds under the assumption of p-integrability of its first weak derivatives in degree p≥n−1. We prove that inverse homeomorphisms have integrable first weak derivatives. For the case p>n we obtain necessary conditions for existence of Sobolev homeomorphisms between manifolds. These necessary conditions based on Poincaré type inequality:

infc∈R∥u−c∣L∞(M)∥≤K∥u∣L1∞(M)∥.

As a corollary we obtain the following geometrical necessary condition:

{\em If there exists a Sobolev homeomorphisms ϕ:M→M′, ϕ∈W1p(M,M′), p>n, J(x,ϕ)≠0 a. e. in M, of compact smooth Riemannian manifold M onto Riemannian manifold M′ then the manifold M′ has finite geodesic diameter.}}

infc∈R∥u−c∣L∞(M)∥≤K∥u∣L1∞(M)∥.

As a corollary we obtain the following geometrical necessary condition:

{\em If there exists a Sobolev homeomorphisms ϕ:M→M′, ϕ∈W1p(M,M′), p>n, J(x,ϕ)≠0 a. e. in M, of compact smooth Riemannian manifold M onto Riemannian manifold M′ then the manifold M′ has finite geodesic diameter.}}

Original language | English |
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Publisher | arXiv:0712.2147 [math.FA] |

State | Published - 2007 |